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Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls

Guillaume Penent, Nicolas Privault

TL;DR

This work proves the existence of continuous viscosity solutions to semilinear elliptic PDEs with fractional Laplacians on open balls, under small exterior data and nonlinear coefficients. It introduces a tree-based probabilistic representation driven by a $(2s)$-stable branching process, yielding the solution as $u(x)=\mathbb{E}[\mathcal{H}(\mathcal{T}_x)]$ and establishing the required integrability and continuity via exit-time regularity. The main contributions include existence results for a broad class of polynomial nonlinearities without degree bounds, with additional results for small radii and finite nonlinearities, and the identification of conditions under which the viscosity solution agrees with a weak solution (uniqueness under $\phi\in H^{2s}$). Numerical experiments up to dimension $d=100$ validate the approach and illustrate its scalability through Monte Carlo tree methods and stable-process simulations. The methods extend nonlocal PDE theory to high dimensions and offer a practical computational framework for semilinear fractional elliptic problems.

Abstract

We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension $d\geq 1$. Our approach relies on a tree-based probabilistic representation based on a (2s)-stable branching processes for all $s\in (0,1)$, and our existence results hold for sufficiently small exterior conditions and nonlinearity coefficients. In comparison with existing approaches, we consider a wide class of polynomial nonlinearities without imposing upper bounds on their maximal degree or number of terms. Numerical illustrations are provided in large dimensions.

Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls

TL;DR

This work proves the existence of continuous viscosity solutions to semilinear elliptic PDEs with fractional Laplacians on open balls, under small exterior data and nonlinear coefficients. It introduces a tree-based probabilistic representation driven by a -stable branching process, yielding the solution as and establishing the required integrability and continuity via exit-time regularity. The main contributions include existence results for a broad class of polynomial nonlinearities without degree bounds, with additional results for small radii and finite nonlinearities, and the identification of conditions under which the viscosity solution agrees with a weak solution (uniqueness under ). Numerical experiments up to dimension validate the approach and illustrate its scalability through Monte Carlo tree methods and stable-process simulations. The methods extend nonlocal PDE theory to high dimensions and offer a practical computational framework for semilinear fractional elliptic problems.

Abstract

We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension . Our approach relies on a tree-based probabilistic representation based on a (2s)-stable branching processes for all , and our existence results hold for sufficiently small exterior conditions and nonlinearity coefficients. In comparison with existing approaches, we consider a wide class of polynomial nonlinearities without imposing upper bounds on their maximal degree or number of terms. Numerical illustrations are provided in large dimensions.

Paper Structure

This paper contains 4 sections, 8 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Probabilistic representation of elliptic PDE solutions
  3. Proofs of existence results
  4. Numerical examples

Key Result

Theorem 1.1

Assume that $|\phi|_{L^\infty (B^c(0,R))} < \infty$ and $\sum_{l\in \mathcal{L}} |c_l|_{L^\infty (B(0,R))} < \infty$ are both sufficiently small. Then, the nonlinear PDE eq:1 admits a (continuous) viscosity solution on $\mathcal{O}=B(0,R)$.

Figures (4)

  • Figure 1: Numerical solutions in dimension $d=1$ with $\alpha =0.8$.
  • Figure 2: Numerical solutions of \ref{['eq:nld0']} in dimension $d=10$ with $\alpha =1.5$.
  • Figure 3: Numerical solutions of \ref{['eq:nld1']} in dimension $d=100$ with $\alpha =1.75$.
  • Figure 4: Numerical solutions of \ref{['eq:nld2']} in dimension $d=10$ with $\alpha =1.75$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Proposition 3.5